Algebraic Topology and Machine Learning

Topological methods and machine learning have long enjoyed fruitful interactions as evidenced by popular algorithms like ISOMAP, LLE and Laplacian Eigenmaps which have been borne out of studying point cloud data through the lens of topology/geometry. More recently several researchers have been attempting to understand the algebraic topological properties of data. Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study and classify topological spaces. The machine learning community thus far has focussed almost exclusively on clustering as the main tool for unsupervised data analysis. Clustering however only scratches the surface, and algebraic topological methods aim at extracting much richer topological information from data.

The goals of our workshop are:
1. To draw the attention of machine learning researchers to a rich and emerging source of interesting and challenging problems.
2. To identify problems of interest to both topologists and machine learning researchers and areas of potential collaboration.
3. To discuss practical methods for implementing topological data analysis methods.
4. To discuss applications of topological data analysis to scientific problems.

We will also target submissions in a variety of areas, at the intersection of algebraic topology and learning, that have witnessed recent activity. Areas of focus for submissions include but are not limited to:
1. Robust geometric inference: New methods to understand the various topological properties of (possibly) noisy random samples, and also the applications of geometric inference techniques to other problems in machine learning (eg. supervised learning).
2. Applications of topological data analysis to new/existing areas: Topological data analysis has previously found application in a variety of areas including medical imaging and neuroscience, sensor networks, landmark-based shape data analyses, proteomics, microarray analysis and cellular biology.
3. Computational aspects of topological inference: Computing topological properties of data often involves building simplicial complexes and computing linear algebraic properties (eg. rank) of matrices associated with these complexes. These methods are typically computationally intensive, and new methods are being continually developed to help scale topological methods to larger datasets.
4. Statistical approaches to topological data analysis: Some recent work in computational topology has focussed on making statistical sense of topological properties. For instance, studying the topological properties of bootstrap samples and using this to associate topological properties of random data with confidence statements typical of statistical hypothesis testing.